Comparison methods for a Keller–Segel-type model of pattern formations with density-suppressed motilities

Global boundedness of solutions to a parabolic-parabolic chemotaxis system with local sensing in higher dimensions

“…This identity is the specific feature of (1.3) differently from the logarithmic Keller-Segel system (1.2). Indeed, in previous results ( [6,7,8]) a priori estimates for w and the comparison estimate…”

“…Concerning solutions to (1.1), there are also many researches corresponding to this conjecture. Boundedness of solutions are established for small k > 0 (for parabolic-parabolic case [4,7,8]; for parabolic-elliptic case [1,8]), but unfortunately in these results the conditions on k is strictly smaller than the conjectured range (0, n n−2 ). In [25] global existence and boundedness are established for any k > 0, but some smallness of another parameter is required, and this smallness condition restricts the size of the initial data (We will see details in Section 6).…”

“…Especially, in the special case where γ(v) = e −v and χ(v) = v, (1.1) and (1.2) have the Lyapunov functionals: [7,13]) [21]), respectively, where…”

“…It is of interest to know whether the solutions of both systems behave similarly or not. In the case where (γ(v), χ(v)) = (e −v , v) and n = 2, by using these Lyapunov functionals and the mass conservation law, global existence and blowup of solutions are studied; Solutions to both systems (1.1) and (1.2) exist globally in time and are uniformly bounded in time if Ω u(x, 0)dx < θ * ( [7,13,21]); there exist unbounded solutions to these systems if Ω u(x, 0)dx > θ * ( [7,13,11]), where θ * = 8π for radially symmetric solutions and θ * = 4π for nonradial solutions. Although (1.2) has finite time blowup solutions ( [19]), solutions of (1.1) exist globally in time and blow up at infinite time ( [7]).…”

Asymptotic behavior of a quasilinear Keller--Segel system with signal-suppressed motility

“…It should be remarked that based on the comparison method, Fujie and Jiang ( [8]) obtained the uniform-in-time boundedness to (1.4) in two-dimensional setting for the more general motility function γ, and in the three-dimensional case under a stronger growth condition on 1/γ respectively. In addition, they investigated the asymptotic behavior to the parabolicelliptic analogue of (1.4) under the assumption max [7,14].…”